Question

A common cylindrical copper wire used in a lab is 841 m long. Find the radius (in mm) of a wire necessary to have 0.5 Ohms of resistance. (The resistivity of copper at room temperature is 1.68×10-8 Ohm × meter). Express the answer (only numerical value) to one decimal place.

Answer 1

The relationship between the resistance R of a wire and its resistivity $\rho$ is given by
$R= \frac{\rho L}{A}$
where L is the length of the wire and A is its cross sectional area.

In the problem, we have $R=0.5 \Omega$, $\rho = 1.68 \cdot 10^{-8} \Omega m$ and $L=841 m$. So we can solve the find the area A:
$A= \frac{\rho L}{R}=2.83 \cdot 10^{-5} m^2$

For a cylindrical wire, the cross sectional area is given by
$A= \pi r^2$
where r is the radius. We know the value of the area A, so now we can find the radius of the wire:
$r= \sqrt{ \frac{A}{\pi} }= \sqrt{ \frac{2.83 \cdot 10^{-5}m^2}{\pi} }=0.003 m=3.0 mm$